Dulyn's Math Mistake: Identifying The Error In Solving For X

Hey guys! Let's dive into a common math problem and see if we can spot a mistake. We're looking at how Dulyn tried to solve for x in the equation 5.4 + x = 12.2. Here's what she did:

5. 4 + x = 12.2 6. 4 - 5.4 x = 6.8

Looks like a pretty straightforward problem, right? But is everything perfect? Our mission, should we choose to accept it, is to figure out if Dulyn made an error and, if so, what it was. We'll break down the steps and use our math knowledge to uncover the truth.

Decoding the Equation and Dulyn's Approach

Alright, let's break down what's happening in the equation. We start with 5.4 + x = 12.2. This is a basic algebraic equation, where we're trying to isolate x (get it by itself) to find its value. The goal is to perform operations on both sides of the equation in a way that keeps things balanced, like a seesaw. If we do something to one side, we must do the same to the other side to keep the equation valid.

Now, let's analyze Dulyn's method. It seems she tried to get rid of the 5.4 on the left side to leave x alone. Her attempt involved subtracting 5.4. However, to correctly isolate x, you have to remember the golden rule of algebra: Whatever you do to one side of the equation, you must do to the other side. This ensures that the equation remains balanced and true.

Dulyn's mistake lies in the execution of this concept. The right way to solve this type of equation is to isolate x, to achieve this, the correct procedure is to subtract 5.4 from both sides. When Dulyn subtracted 5.4 only from the left side, it led to an incorrect answer. It's like trying to bake a cake and only adding the eggs to half the batter – you won't get a proper cake! Understanding this balance is the core of algebra and avoiding errors in solving for x.

Identifying the Error: A Closer Look

So, what exactly went wrong? Let's zoom in on Dulyn's steps:

1. 5.4 + x = 12.2
2. 5.4 - 5.4
3. x = 6.8

Looking at this, it's clear that Dulyn understood the need to remove the 5.4 from the left side. However, she didn't apply the operation correctly. The correct method involves subtracting 5.4 from both sides of the equation. This maintains the balance, ensuring that we get the right answer.

The Crucial Error: Dulyn only subtracted 5.4 on the left side, but she should have also subtracted it from the right side (12.2). This is where the balance of the equation was lost. The correct steps would look like this:

1. 5.4 + x = 12.2
2. 5.4 + x - 5.4 = 12.2 - 5.4 (Subtracting 5.4 from both sides)
3. x = 6.8

By subtracting 5.4 from both sides, we isolate x and arrive at the correct answer.

The Correct Approach: Solving for x

Let's walk through the right way to solve the equation 5.4 + x = 12.2. Remember, the key is to isolate x while keeping the equation balanced. Here’s a step-by-step guide:

1. Identify the Constant: The constant is the number that is added to or subtracted from x. In our case, it's 5.4.
2. Isolate x: To isolate x, we need to get rid of the 5.4 on the left side. The opposite of adding 5.4 is subtracting 5.4. So, we subtract 5.4 from both sides of the equation.
3. Perform the Operation: 5.4 + x - 5.4 = 12.2 - 5.4. On the left side, 5.4 - 5.4 cancels out, leaving us with just x. On the right side, 12.2 - 5.4 equals 6.8.
4. The Solution: After these steps, the equation becomes x = 6.8. And that, my friends, is the correct solution.

By consistently applying the same operation to both sides of the equation, we maintain balance, ensuring an accurate result. This is a fundamental concept in algebra.

The Answer Choices and Why They Matter

Now, let's see how this ties into the multiple-choice options, which help to clarify our understanding of the error.

A. She should have added 5.4 to each side. This is incorrect. To isolate x, you need to subtract 5.4 from both sides, not add it. Adding 5.4 to both sides would make the equation even more complex and move us further from the solution. B. She should have added the inverse of 5.4 to each side. This is the correct answer. The inverse of 5.4 in this context is -5.4 (because 5.4 + (-5.4) = 0). So, Dulyn should have subtracted 5.4 from both sides. This is the same as adding the inverse of 5.4. This is the same as the correct answer. C. She made a calculation error. While a calculation error occurred, the core of the problem is a conceptual misunderstanding of algebraic balance. The calculation error stemmed from not applying the operation to both sides, which is a conceptual problem.

The Correct Answer Explained: The best answer is B. She should have added the inverse of 5.4 to each side. This highlights the understanding that solving an equation involves performing the same operation (in this case, subtracting 5.4, which is equivalent to adding the inverse, -5.4) on both sides to maintain balance and correctly isolate x.

Key Takeaways: Mastering Equation Solving

So, what have we learned from Dulyn's little math adventure? Here are the most important takeaways:

• Balance is Key: In algebra, always remember the balance. Whatever you do to one side of the equation, you must do to the other side.
• Isolate the Variable: The goal is to get the variable (in this case, x) by itself on one side of the equation.
• Inverse Operations: Use inverse operations to isolate the variable. The inverse operation is the one that undoes the original operation. For example, the inverse of addition is subtraction, and the inverse of multiplication is division.
• Practice Makes Perfect: The more you practice solving equations, the better you'll become at recognizing the correct steps and avoiding common mistakes.

By understanding these principles, you'll be well-equipped to tackle various algebraic problems, ensuring you get the correct answers and building a strong foundation in mathematics. Keep practicing, keep learning, and don't be afraid to make mistakes – that's how we all grow!